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8.4 The Discriminate

Need To Know
▪ The Discriminate
▪ Predicting roots
▪ Writing equations from solutions

For ax2 + bx + c = 0, The Discriminate is ________________

The Discriminate

 Discriminate Nature of Solutions 0 Positive – Perfect Squar Positive – Not a perfect Negative

Predicting the Solution Type

Determine what type of solution each has.

x2 – 7x + 5 = 0

x2 + 4x + 6 = 0

9t2 – 48t + 64 = 0

end

Need To Know
▪ Finding the vertex and axis of symmetry
▪ Vertex Form of an equation

Graph f (x) = ax2: The effect of “a”?  Label vertex point, use axis of sym.

Observation:
a > 1, then parabola is _________
a < 1, then parabola is _________
a < 0, then parabola is _________

The Graph of f (x) = a(x – h)2

Graph:  Label vertex point, use axis of sym.

Observation:
(x+h) h is neg, then parabola _________
(x–h) h is pos, then parabola _________
The effect of h is _________

The Graph of f (x) = a(x – h)2 + k

Graph:  Label vertex point, use axis of sym.

Observation:
The vertex is _________
The equation of the A.O.S. _________
The effect of k _________

Graphing y = a(x – h)2 + k

Graph:  Summarize the Vertex Form

y = a(x – h)2 + k is the Vertex Form.

Three parameters: a, h, k

Maximum or Minimum

end

Need To Know
▪ Review completing the square
▪ Converting into Vertex Form
(not to learn but to appreciate the short cut formula)
▪ Vertex Point Formula – Short Cut
▪ Finding intercepts
▪ Sketching the graph of a quadratic

Completing the Square

Recall:

x2 – 10x + _______ =

Recall:

y = a(x – h)2 + k
What is the vertex point? ____________

If the quadratic is in standard form, we have no information.
We need to change the form into vertex form (squared stuff).

g(x) = x2 – 10x + 21
f(x) = 4x2 + 8x - 3

Vertex Point – short cut (easy way)

If f(x) = ax2 + bx + c, then (h, k) = ___________
which means _____________________________.
1. Find h with the formula
2. Find k by plugging h into the function.

g(x) = x2 – 10x + 21
f(x) = 4x2 + 8x – 3

Intercept Points

Recall:
X-intercept point = point where graph crosses x-axis. (a, 0)
Find the x-intercept point by letting y = 0.

Y-intercept point = point where graph crosses y-axis. (0, b)
Find the y-intercept point by letting x = 0.
Example:

Find the intercepts of f(x) = x2 + 5x – 6

Sketch Graph

y = x2 + 6x + 5
Label vertex point,
Label intercept points Sketch Graph

y = -3x2 + 6x + 2

Label vertex point,
Label intercept points end